1 00:00:00,012 --> 00:00:06,900 We're now going to look at a different category of games, games that are called 2 00:00:07,262 --> 00:00:15,250 Bayesian games. Sometimes called games of incomplete information, not to be confused 3 00:00:15,262 --> 00:00:22,280 with games of imperfect information. So far what we have seen are games in which 4 00:00:22,292 --> 00:00:28,340 all agents know what the basic setting is. That is, they know who the players are. 5 00:00:28,462 --> 00:00:34,665 They know the actions available to the players. They know that payoffs associated 6 00:00:34,677 --> 00:00:41,768 with each strategy profile or each action profile, depending on what everybody does. 7 00:00:42,147 --> 00:00:47,572 this is true in all games, including games of imperfect information. That is games 8 00:00:47,584 --> 00:00:53,905 of, in which our information is such where agents don't know. Exactly in which state 9 00:00:53,917 --> 00:01:01,290 they are, nonetheless they know what would happen given what the strategy of all the 10 00:01:01,302 --> 00:01:07,638 agents. So we're going to relax that. We're going to assume that What you said 11 00:01:07,650 --> 00:01:12,562 isn't necessarily always common knowledge. Now in principle you can imagine relaxing 12 00:01:12,574 --> 00:01:17,246 the various assumptions. You don't know the number of the players. You don't know 13 00:01:17,497 --> 00:01:23,117 maybe the action, how many actions are available to them. >> But, in some sense, 14 00:01:23,243 --> 00:01:30,089 some informal sense, all of those forms of uncertainties can be reduced to one type 15 00:01:30,101 --> 00:01:36,844 of uncertainty, that is about the payoffs in the game. And, so we will assume that 16 00:01:36,856 --> 00:01:43,424 agents have perfect common knowledge of everything, except what the payoffs of the 17 00:01:43,436 --> 00:01:50,240 game are. And, furthermore, that there is some. Prior knowledge, prior belief that 18 00:01:50,252 --> 00:01:56,055 is common to all the agents about those payoffs, and simply agents have different 19 00:01:56,067 --> 00:02:02,309 signals that lead to different posteriors based on those common prior This may sound 20 00:02:02,321 --> 00:02:09,807 very vague. Let me make it precise. Let me first give the formal definition. and then 21 00:02:09,819 --> 00:02:15,520 just give an example which will make everything clear. So we have a set of 22 00:02:15,532 --> 00:02:22,395 games, that is a Bayesian game is defined by first of all a set of games that are 23 00:02:22,407 --> 00:02:28,120 identical except in their payoffs. So let's start going over the formal 24 00:02:28,132 --> 00:02:34,881 definitions again. So we have A tuple that defined the game. You have a set of agents 25 00:02:34,893 --> 00:02:40,924 N, and we have G as a set of regul ar kind of games. Think of these as normal for 26 00:02:40,936 --> 00:02:47,501 games for example. each game is a consists of N A agent play the game, and they all 27 00:02:47,513 --> 00:02:53,647 have the same strategy space. That is, they're 2 games in the set that have the 28 00:02:53,659 --> 00:03:01,855 same strategy space. As I said, the payoff will be generally different. We have a 29 00:03:01,867 --> 00:03:08,550 prior that is a distribution over those set of games. That's a sum prior. Nature 30 00:03:08,562 --> 00:03:15,205 will decide which game is actually played based on this prior. And then there's 31 00:03:15,217 --> 00:03:21,430 private signals as defined by our partition structure. that is each agent 32 00:03:21,762 --> 00:03:28,115 for each agent, we find some equivalent relation on the games. And agent will be, 33 00:03:28,242 --> 00:03:35,940 sort of, told which in which equivalent's class they are. And based on that, they'll 34 00:03:35,952 --> 00:03:42,149 need to play the game. Now this is a mouth, mouthful, I know but hopefully the 35 00:03:42,161 --> 00:03:47,645 following example will make it clear. Let's assume that we have 4 possible 36 00:03:47,657 --> 00:03:53,108 games. And here the games that are familiar, we have Matching Pennies, we 37 00:03:53,120 --> 00:03:58,903 have Prisoners Dilemma, we have the game of pure Coordination, and we have Bat, 38 00:03:59,023 --> 00:04:06,070 Battle of teh Sexes. Each of those defined simply by their, by their payoffs. Now, 39 00:04:06,537 --> 00:04:13,385 nature is going to decide which of those games actually is being played. And we've 40 00:04:13,397 --> 00:04:20,150 decided, based on the probabilities as listed here. We have a probability of 0.3 41 00:04:20,162 --> 00:04:25,926 here, 0.1 here, 0.2 here, and Point 4 here. Now once nature makes its choice, 42 00:04:26,040 --> 00:04:31,739 agent will play. But the question is, what will they know? They will know the prior, 43 00:04:31,853 --> 00:04:37,579 but they will know something in addition. And what they will know will be defined by 44 00:04:37,591 --> 00:04:42,968 this partition. So here we have the 2 agents playing, and for each of these 45 00:04:42,980 --> 00:04:46,960 agents there is a, a, an equivalent to find. 46 00:04:46,960 --> 00:04:51,951 So, for example, think about the role player. For the role player there are 2 47 00:04:51,963 --> 00:04:56,016 equivalence classes, denoted by the bold Partition. 48 00:04:56,022 --> 00:05:03,950 And I'll make it green now. This is the equivalent relation defined for the row 49 00:05:03,962 --> 00:05:11,877 agent. So, for example, suppose that, nature decided to, in fact. Playing 50 00:05:11,889 --> 00:05:18,089 matching pennies. The agents will know the, that is the row agent, will know that 51 00:05:18,101 --> 00:05:24,093 he's either in this game, or in this game. He'll know that he's not in any of the 52 00:05:24,105 --> 00:05:30,255 other games. So that will be his private signal. He'll now have posterior belief. 53 00:05:30,378 --> 00:05:38,710 What will he believe? Well he will believe that with probability point a, point 75 he 54 00:05:38,722 --> 00:05:46,195 is playing this game at 0.25 he is playing this game and why is that because this is 55 00:05:46,207 --> 00:05:52,840 the ratio of 3 to 1 as defined between these 2 games for him. What will the 56 00:05:52,852 --> 00:06:00,790 column player Now well a column player lets pick a different color for her, she 57 00:06:00,802 --> 00:06:07,985 has a different equal ventilation, this one. And now if matter again chose 58 00:06:07,997 --> 00:06:15,652 matching penny what will she know? Well she'll know that she is either. In this 59 00:06:15,664 --> 00:06:23,799 game or in this game, and in this case she will need to update her prior to reflect 60 00:06:23,811 --> 00:06:31,129 this information and the perceiver for the a,h column agent will be that she is 61 00:06:31,141 --> 00:06:37,910 playing this game's probability .6 and this Proba, probability 0.4. Again, 62 00:06:38,042 --> 00:06:43,960 maintaining the ratios between these 2 games. And then we'll know more, 63 00:06:44,092 --> 00:06:51,780 intuitively. Because the, when the agent knows, The ag, for example, the row agent 64 00:06:51,792 --> 00:06:58,695 knows that she's someplace in this class. She will not know exactly what information 65 00:06:58,707 --> 00:07:05,730 the common player has, but she knows what the possible information is it might have. 66 00:07:05,857 --> 00:07:14,528 She knows that, that the role player knows that Either she is in this game, in which 67 00:07:14,540 --> 00:07:23,927 case she knows that this would be the information that the common player has or 68 00:07:23,939 --> 00:07:30,060 that she is in this game in which case she, the role player knows that the role 69 00:07:30,072 --> 00:07:36,507 player knows that she's someplace here. And so it's a complicated story because 70 00:07:36,519 --> 00:07:43,277 you can keep going. They have some beliefs about what the other player believe about 71 00:07:43,289 --> 00:07:50,102 what they know so on and so forth. But this is the structure of Bayesian games 72 00:07:50,114 --> 00:07:57,796 and based on this, you can start modeling and what it will do. But since this is 73 00:07:57,808 --> 00:08:06,667 complicated there's an alternative Perspective on, on, on beige in game that 74 00:08:06,679 --> 00:08:12,486 is different, but in some sense easier to work with.